On a local solvability of the contact Muskat problem

TL;DR

This paper proves local classical solvability for a 2D contact Muskat problem with free boundary and describes conditions for the occurrence of a waiting time phenomenon, advancing understanding of interface evolution in fluid dynamics.

Contribution

It establishes the one-to-one local classical solvability of the contact Muskat problem with specific boundary conditions and initial interface shapes.

Findings

Proves local classical solvability under certain conditions.

Identifies sufficient conditions for the waiting time phenomenon.

Analyzes the influence of initial interface shape on solution behavior.

Abstract

In the paper, we discuss the two-dimensional contact Muskat problem with zero surface tension of a free boundary. The initial shape of the unknown interface is a smooth simple curve which forms acute corners $\delta_{0}$ and $\delta_{1}$ with fixed boundaries. Under suitable assumptions on the given data, the one-to-one local classical solvability of this problem is proved. We also describe the sufficient conditions on the data in the model which provide the existence of the "waiting time" phenomenon.